Question
Two arithmetic progression have the same common difference. The difference between their $100^{\text {th }}$ terms is $100$ , What is the difference between their $1000^{\text {th }}$ terms?
✓
Answer
Let the two A.P. is be $a_1, a_2, a_3, \ldots .$. and $b_1, b_2, b 3, \ldots .$.
$a_n=a_1+(n-1) d \text { and } b_n=b_1+(n-1) d$
Since common difference of two equations is same given difference between $100^{\text {th }}$ terms is $100$
$a_{100}-b_{100}=100$
$a_1+(99) d-b_1-99 d=100$
$a_1-b_1=100 \ldots . . .(i)$
Difference between. $1000th$ terms is
$a_{1000}-b_{1000}=a_1+(1000-1) d-\left(b_1+(1000-1) d\right)$
$=a_1+999 d-b_1-999 d$
$=a_1-b_1$
$=100(\text { from }(1))$
$\therefore$ Hence difference between $1000^{\text {th }}$ terms of two A.P. is $100 .$
$a_n=a_1+(n-1) d \text { and } b_n=b_1+(n-1) d$
Since common difference of two equations is same given difference between $100^{\text {th }}$ terms is $100$
$a_{100}-b_{100}=100$
$a_1+(99) d-b_1-99 d=100$
$a_1-b_1=100 \ldots . . .(i)$
Difference between. $1000th$ terms is
$a_{1000}-b_{1000}=a_1+(1000-1) d-\left(b_1+(1000-1) d\right)$
$=a_1+999 d-b_1-999 d$
$=a_1-b_1$
$=100(\text { from }(1))$
$\therefore$ Hence difference between $1000^{\text {th }}$ terms of two A.P. is $100 .$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
The monthly income of $100$ families are given as below:
Calculate the modal income.
→|
Income in ₹
|
Number of families
|
|
$0-5000$
|
$8$
|
|
$5000-10000$
|
$26$
|
|
$10000-15000$
|
$41$
|
|
$15000-20000$
|
$16$
|
|
$20000-25000$
|
$3$
|
|
$25000-30000$
|
$3$
|
|
$30000-35000$
|
$2$
|
|
$35000-40000$
|
$1$
|
The areas of two similar triangles are $100\ cm^2$ and $49\ cm^2$ respectively. If the altitude of the bigger triangle is $5\ cm$, find the corresponding altitude of the other.
→In the given figure, the side of square is $28\ cm$ and radius of each circle is half of the length of the side of the square where O and O' are centres of the circles. Find the area of shaded region.
→Find n if the given value of $x$ is the $n^{th}$ term of the given A.P.
$5\frac{1}{2},11,16\frac{1}{2},22,\ .....;\text{ x}=550$
→$5\frac{1}{2},11,16\frac{1}{2},22,\ .....;\text{ x}=550$
Solve the following simultaneous equations using Cramer’s rule.
x + 2y = –1; 2x – 3y = 12
x + 2y = –1; 2x – 3y = 12
Find:$9^{th}$ term of the A.P. $\frac{3}{4},\frac{5}{4},\frac{7}{4},\frac{9}{4}, .....$
→Tickets numbered 1 to 30 are mixed up together and then a ticket is drawn at random. What is the probability that the ticket drawn will be a multiple of 7 ?
In △ ABC point △ on side BC is such that △C = 6, BC = 15. Find A(△ ABD) : A(△ ABC) and A(△ ABD) : A(△ ADC) .
Find the roots of the following equations, if they exist, by applying the quadratic formula:$3\text{x}^2-2\sqrt6\text{x}+2=0$
→Solve the following equations by using the method of completing the square:
$x^2 - 4x + 1 = 0$
→$x^2 - 4x + 1 = 0$